In this work, we investigate the complete convergence of randomly weighted sums of mixing random variables. Using the theory of regularly varying functions, we present results that are more general than previous works. Let \(\{X_n; n\ge 1\}\) be a sequence of \(\rho ^*\) -mixing random variables, and \(\{A_{n i}, 1 \le i \le n\}\) be a triangular array of \(\rho ^*\) -mixing random variables that are independent of \(\{X_{n};n\ge 1\}\) . Under a mild condition, we give the necessary and sufficient conditions for the convergence of the series \(\begin{aligned} \sum _{n=1}^{\infty }\frac{f(n)}{n^{2}}\mathbb {P}\left( \max \limits _{1 \le k \le n} \left| \sum _{i=1}^{k}A_{ni}X_{i} \right|>\varepsilon g(n)\right) \text { for all }\epsilon >0, \end{aligned}\) where f and g are regularly varying functions. Moreover, we also consequently give a result for asymptotics of the quasi-renewal counting process. The main results are subsequently applied to simple linear regression models and state observers of linear-time invariant systems. Examples and numerical simulations are also provided to demonstrate our results.
